Wierd faces after I moved one vertex of quads
Result comes from fact that by draging only one vertex of a quad face makes face non-planar, in viewport hardly seen as two triangles (there is not any other mathematical way to result). In real life if you take plate of glass and you would take one corner lower you would have to split plate by diagonal cut between any two corners. 

In your case is one face triangulated in shorter (red) direction and second in longer (green).





Yet not sure how final surface should looks like for you, but you can create like negative shape from two triangles and "cut" it out by Boolean modifier > Difference from your base mesh.

Create a new object with two triangle faces. Like that you keep faces planar.



For Boolean modifier you will need a non-manifold mesh. So extrude it by Solidify modifier to create one close mesh. It is non destructive way. So you can go to edit mode and adjust the "cut" shape anytime.

Now use Boolean modifier for your base object to Difference it.
Also I changed the viewport display mode for this object to "Wire" so you can see how Boolean modifer perform on base mesh. 





Note: Blender has also an operator, you can search Make Planar Faces, that can recalculate them into flat faces, but in cost of distortions of all selected vertices.